Spontaneous Symmetry Breaking in Non Abelian Gauge Theories Michael LeBlanc Abstract This term paper explores the phenomenon of spontaneous symmetry breaking in systems with non-abelian gauge symmetry. The most famous example comes from the standard model of particle physics, wherein symmetry breaking is responsible for the masses of the W and Z bosons and the corresponding short range of the weak nuclear force. The electroweak theory will be discussed as well as a more ambitious (but less successful) attempt to unify the strong, weak and electromagnetic forces in which symmetry breaking also plays a key role. 1 Introduction The history of the principle of gauge invariance [9] is a long and storied one that stretches far back into the nineteenth century, beginning with the observation that the electromagnetic scalar and vector potential were underdetermined and thus could be subjected to somewhat arbitrary additional constraint equations (gauge conditions). In 1926, Fock discovered that a charged particle’s wave equation was invariant under a position and time-dependent phase transformation φ exp(iθ(x , t ))φ provided that → the electromagnetic potentials were properly adjusted by a gauge transformation to compensate. The idea that the electromagnetic coupling can be looked at as a conse- quence of local phase invariance, which traces back to a speculation made by Weyl in 1928, stands at the center of particle physics today. In the mid fifties, Yang and Mills [16] showed that the gauge principle could be generalized from phase (U(1)) to isospin (SU(2)) transformations. The main difficulty associated with this extension is that the isospin transformations do not commute with one another, thus their theory is a termed non-abelian gauge theory, in contrast with the abelian electromagnetism. Their approach is easily generalized from SU(2) to any compact lie group, therefore gauge theories have the allure of associating to an abstract symmetry group of one’s choosing a unique theory of interacting matter and gauge fields. Notwithstanding this allure, these theories at first seemed unsuitable for describing fundamental interactions since they involved massless gauge bosons which had not been observed and were deemed unfit for describing the short-ranged nuclear forces. It turns out this problem can be avoided in two ways: the bosons can become massive due 1 to spontaneous symmetry breaking or the bosons can not be observed in the particle spectrum due to confinement. Both of these possibilities are realized in the weak and the strong force, respectively, but it is the first that will be the main subject of this paper. The outline is as follows: First, I will give a brief mathematical introduction to theories with spontaneously broken continuous symmetries, beginning with Goldstone’s U(1) model and the abelian Higgs model. This is really nothing more than the Ginzburg-Landau theory learned in class, but it will give me a chance to establish relativistic notation in a familiar context. Then I will generalize to non abelian gauge theories, keeping the mathematical overhead to a minimum. After the formalism is established I will explore three different theories with different symmetry breaking patterns. The first is SU(2) U(1), which is a simple model for → electroweak symmetry breaking with no Z boson. This model is not correct, but it is still interesting. For instance, it has magnetic monopole solutions which are an example of topological defects as a consequence of symmetry breaking. The analogue from class would be flux tubes in the U(1) theory. The second example is SU(2) U(1) U(1) . This is the celebrated Glashow- × Y → EM Weinberg-Salam model. I will discuss matter content and the experimental successes of this model. I will also briefly discuss some of its shortcomings as well as alternative symmetry breaking mechanisms to the standard model Higgs. The third example is SU(5) SU(3) SU(2) U(1). This is Georgi and Glashow’s → × × minimal Grand Unified Theory (GUT) attempting to unify the strong, weak and elec- tromagnetic forces. 2 Symmetry Breaking in Gauge Theories 2.1 U(1) theory For our first example of a theory with a continuous symmetry [5], we consider a complex scalar field φ with relativistic action S = d4x where L µ 2 =(∂ φ)∗∂ φ V ( φ ). L µ − | | The action is invariant under transformations of the form φ eiθφ, so we say the → theory has internal U(1) symmetry. The equation of motion (Euler-Lagrange equation) derived from this action is ∂V ∂2φ + φ =0, ∂ φ 2 | | which becomes the nice linear Klein Gordon equation for V = m2 φ 2. The Hamiltonian | | is H = d3x ∂ φ 2 + φ 2 + V ( φ 2). | t | |∇ | | | Notice that the lowest energy field configuration is a constant that minimizes V . To establish a symmetry breaking minimum, we select a quartic potential V = λ ( φ 2 v2/2)2 where v is a real number. The minimum field configuration then 2 | | − 2 2 becomes the degenerate circle φ 2 = v . We can then expand the Lagrangian about | | 2 the arbitrarily selected minimum φ0 = v/√2 in terms of new fields η(x) and θ(x)where φ(x)= 1 (v + η(x))eiθ(x)/v and obtain √2 1 (1 + η/v)2 λv2 = ∂ η∂µη + ∂ θ∂µθ η2 + O(η3). L 2 µ 2 µ − 2 Our new fields obey linearized equations of motion 2 2 2 (∂ + mη)η = 0; ∂ θ = 0 (exact) 2 2 where mη = λv and the θ field is massless, the first example of a Goldstone mode. Now, return to the original version of the Lagrangian and consider local U(1) trans- formations of the form φ eiθ(x)φ where the transformation parameter is a function → of space-time. Now, the derivative terms in the action are not invariant, but we can compensate by replacing the derivatives with covariant derivatives D ∂ ieA µ ≡ µ − µ and having the vector field A compensate by transforming A A + 1 ∂ θ. This is µ µ → µ e µ an example of the principle mentioned above: a global symmetry can be promoted to a local symmetry with the addition of a vector field and the form of the interactions are determined by this process. Notice that the transformation on Aµ is just a gauge trans- formation from electricity and magnetism, so we’ll identify it with an electromagnetic field and write the Lagrangian: µ 2 1 µν =(D φ)∗D φ V ( φ ) F F L µ − | | − 4 µν where F ∂ A ∂ A is the electromagnetic field tensor, and the last term was µν ≡ µ ν − ν µ chosen to generate Maxwell’s vacuum equations when φ =0. Notice that the term (and in fact Fµν itself) is gauge invariant, so its addition to the Lagrangian does not break the local symmetry. We’re in position to see the abelian Higgs mechanism first derived in this form in [6]. Consider the same potential as before and expand again about the same φ0, we get 1 (1 + η/v)2 m2 1 = ∂µη∂ η + (∂µθ evAµ)(∂ θ evA ) η η2 F µνF + O(η3). L 2 µ 2 − µ − µ − 2 − 4 µν We can perform a gauge transformation A A + 1 ∂ θ and obtain µ → µ ev µ 1 m2 v2e2 1 = ∂µη∂ η η η2 + AµA F µνF + higher order terms. L 2 µ − 2 2 µ − 4 µν µν The linearized equation of motion for the electromagnetic field is therefore ∂µF = e2v2Aν. The Lorenz gauge condition ∂ Aµ =0, which can be assumed from the − µ outset, is not ruined by the gauge transformation since ∂2θ =0. Therefore the equation of motion becomes 2 2 µ (∂ + mγ)A =0 2 2 2 where mγ = e v . The electromagnetic field has obtained a mass. 3 We’ve put the Higgs mechanism we learned about in class into the language of relativistic field theory. The main idea is that since a Goldstone boson in a gauge theory is an excitation along an unphysical degree of freedom, it cannot correspond to a physical particle. However, removing the boson from the theory amounts to a choice of gauge. We’ve now used up our option to choose a gauge that’s normally used to make the polarization transverse; the longitudinal polarization is now a physical degree of freedom and the gauge boson can have a mass. This means we are left with a massive η particle (Higgs), a massive gauge boson and no Goldstone boson. 2.2 Yang-Mills Theory We will now generalize to scalar theories invariant under SU(N) transformations φ → exp(iθaτa)φ [16] Here, φ =(φ1,...φN ) is a complex N-tuple and τa are generator matrices, which act as basis vectors for the SU(N) at the identity. In the most common 1 SU(2) case, the index a goes from one to three and τa = 2 σa where σa are the Pauli 1 matrices. The generators can be chosen [2] to obey tr(τa) = 0, tr(τaτb)= 2 δab and [τa,τb]=ifabcτc where the fabc is a real, antisymmetric tensor. In the familiar SU(2) case, fabc = abc. We can take a similar Lagrangian for the scalar field as above = ∂φ 2 V ( φ 2) L | | − | | (where Lorentz and SU(N) index sums are implied) which has the required invariance. To promote the symmetry to a local symmetry, we must replace the derivatives with covariant derivatives D = ∂ igA . The gauge field must transform like A µ µ − µ µ → exp(iθ τ )(A + 1 ∂ θ τ )exp( iθ τ ) so that D φ exp(iθ τ )D φ.